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Local convergence of mean-field Langevin dynamics: from gradient flows to linearly monotone games

Guillaume Wang, Lénaïc Chizat

TL;DR

This work analyzes the local convergence of diffusive mean-field systems, including Wasserstein gradient flows, MFLD, MFL-DA, and multi-species flows, near equilibrium. By assuming a Poincaré inequality for the stationary measure and a linear monotonicity condition on the velocity field, the authors establish exponential local convergence in $\chi^2$-divergence with sharp rates, without requiring displacement convexity. The core technique blends a Hessian-based Otto-calculus viewpoint with a Lyapunov functional that couples $\chi^2$-divergence and weighted $H^{-1}$ norms, yielding explicit rates and neighborhood sizes; the framework also extends to non-gradient, linearly monotone games and polymatrix continuous games, with analogous local rates. The results bridge known global convergence results under strong convexity with new local stability guarantees in broader settings and offer a spectral-theoretic interpretation of the exact asymptotic rate via a Hessian-inspired operator. Open questions include a full spectral characterization of the exact rate and global convergence in nonconvex-nonconcave regimes.

Abstract

We study the local convergence of diffusive mean-field systems, including Wasserstein gradient flows, min-max dynamics, and multi-species games. We establish exponential local convergence in $χ^2$-divergence with sharp rates, under two main assumptions: (i) the stationary measures satisfy a Poincaré inequality, and (ii) the velocity field satisfies a monotonicity condition, which reduces to linear convexity of the objective in the gradient flow case. We do not assume any form of displacement convexity or displacement monotonicity. In the gradient flow case, global exponential convergence is already known under our linear convexity assumption, with an asymptotic rate governed by the log-Sobolev constant of the stationary measure. Our contribution in this setting is to identify the sharp rate near equilibrium governed instead by the Poincaré constant. This rate coincides with the one suggested by Otto calculus (i.e. by a tight positivity estimate of the Wasserstein Hessian), and refines some results of Tamura (1984), extending them beyond quadratic objectives. More importantly, our proof technique extends to certain non-gradient systems, such as linearly monotone two-player and multi-player games. In this case, we obtain explicit local exponential convergence rates in $χ^2$-divergence, thereby partially answering the open question raised by the authors at COLT 2024. While that question concerns global convergence (which remains open), even local convergence results were previously unavailable. At the heart of our analysis is the design of a Lyapunov functional that mixes the $χ^2$-divergence with weighted negative Sobolev norms of the density relative to equilibrium.

Local convergence of mean-field Langevin dynamics: from gradient flows to linearly monotone games

TL;DR

This work analyzes the local convergence of diffusive mean-field systems, including Wasserstein gradient flows, MFLD, MFL-DA, and multi-species flows, near equilibrium. By assuming a Poincaré inequality for the stationary measure and a linear monotonicity condition on the velocity field, the authors establish exponential local convergence in -divergence with sharp rates, without requiring displacement convexity. The core technique blends a Hessian-based Otto-calculus viewpoint with a Lyapunov functional that couples -divergence and weighted norms, yielding explicit rates and neighborhood sizes; the framework also extends to non-gradient, linearly monotone games and polymatrix continuous games, with analogous local rates. The results bridge known global convergence results under strong convexity with new local stability guarantees in broader settings and offer a spectral-theoretic interpretation of the exact asymptotic rate via a Hessian-inspired operator. Open questions include a full spectral characterization of the exact rate and global convergence in nonconvex-nonconcave regimes.

Abstract

We study the local convergence of diffusive mean-field systems, including Wasserstein gradient flows, min-max dynamics, and multi-species games. We establish exponential local convergence in -divergence with sharp rates, under two main assumptions: (i) the stationary measures satisfy a Poincaré inequality, and (ii) the velocity field satisfies a monotonicity condition, which reduces to linear convexity of the objective in the gradient flow case. We do not assume any form of displacement convexity or displacement monotonicity. In the gradient flow case, global exponential convergence is already known under our linear convexity assumption, with an asymptotic rate governed by the log-Sobolev constant of the stationary measure. Our contribution in this setting is to identify the sharp rate near equilibrium governed instead by the Poincaré constant. This rate coincides with the one suggested by Otto calculus (i.e. by a tight positivity estimate of the Wasserstein Hessian), and refines some results of Tamura (1984), extending them beyond quadratic objectives. More importantly, our proof technique extends to certain non-gradient systems, such as linearly monotone two-player and multi-player games. In this case, we obtain explicit local exponential convergence rates in -divergence, thereby partially answering the open question raised by the authors at COLT 2024. While that question concerns global convergence (which remains open), even local convergence results were previously unavailable. At the heart of our analysis is the design of a Lyapunov functional that mixes the -divergence with weighted negative Sobolev norms of the density relative to equilibrium.
Paper Structure (32 sections, 23 theorems, 154 equations)

This paper contains 32 sections, 23 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Related works
  3. Convergence analyses of MFLD.
  4. The mean-field Langevin descent-ascent (MFL-DA) dynamics.
  5. Multi-species flows.
  6. Concurrent work.
  7. Notations
  8. Function spaces.
  9. Functional inequalities.
  10. Heuristics and proof ideas
  11. Heuristics
  12. Proof idea: tracking the H-1 norm
  13. Local L2 convergence of mean-field Langevin dynamics
  14. Local L2 convergence
  15. Tightness of the rate for overdamped Langevin dynamics
  16. ...and 17 more sections

Key Result

Theorem 1.1

Suppose that $F$ is displacement-smooth and linearly convex and that MFLD admits a stationary measure $\nu$. If $\nu$ satisfies PI with a constant $c_{\mathrm{PI}}$, then MFLD converges locally at an exponential rate of at least $2 \tau c_{\mathrm{PI}}$ in $\chi^2\left( \cdot \middle\| \nu \right)$.

Theorems & Definitions (53)

  • Theorem 1.1: Local $\chi^2$ convergence of MFLD, informal
  • Theorem 1.2: Local $\chi^2$ convergence of MFL-DA, informal
  • Theorem 1.3: Local $\chi^2$ convergence of multi-species flows with diffusion, informal
  • Definition 1.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.1
  • ...and 43 more